Spatial embedding and inherited metric constraints are a fundamental trait of
biological neuronal circuits. However their role in shaping connectivity and
dynamics has been often disregarded, with models of neuronal networks paying
much more attention to the distribution of connections in the quest for
understanding network's behavior. In this thesis we aim at filling this gap by

studying the importance of metric features in the complex connectivity-
dynamics-noise interplay that shapes spontaneous neuronal activity.

This thesis combines experiments in rat dissociated neuronal cultures with
theoretical analyses to better comprehend the relevance of spatial embedding.
We developed a new theoretical model grounded on Ising Models to assess
metric effects in neuronal cultures' behavior, and in the context of percolation
approaches. Once metric effects were settled, we illustrated their relevance in
shaping spontaneous activity by perturbing the structural connectivity blueprint
of neuronal cultures. This was achieved by patterning the substrate where
neurons grow, and by using topographical molds that dictated the connectivity
of the network. Next, and since the initiation of bursting activity is governed in
great manner by a complex amplification mechanism that involves metric
correlations and noise, we focused on the metric-driven amplification of
spontaneous single-neuron noise to derive an analytical model that predicts the
frequency of bursting events in neuronal cultures. We then further investigated
in an experimental context the contribution of noise to the observed activity
patterns, and by implementing a moderate electrical stimulation protocol that
increases the level of activity noise in cultures. Finally, the latter study was
completed with experiments regarding the specific role of inhibition in neuronal
networks, to provide a wider understanding of the mechanisms that govern the
initiation and propagation of activity fronts in cortical cultures.

The research of laws able to describe behavioral regularities found in human society has a long history. The pioneering work can be attributed to the mathematician philosopher Thomas Hobbes, who, in 1651, wrote Leviathan, in which he started to create the theo- retical framework of universal features regarding individuals’ preferences and the nature of the interaction between them. Thomas Hobbes theory was the first step in trying to understand the fundamental rules governing social systems and to make predictions on collective behaviors [1]. This intuition about humans’ collective behavior is illustrated on Leviathan’s famous cover shown in Fig 1.1, where the State is represented as a single body made up of individuals

"the people is not in being before the constitution of government as not being any person, but a multitude of single persons" [2].

In this thesis, we have studied mechanical aspects of some biological processes in cells and tissues, which we addressed by developing theoretical models based on the physics of soft active matter. The thesis contains three parts that focus on different biological systems. In Part I, we study the adhesion between the plasma membrane and the actin cortex of eukaryotic cells. We propose a continuum model for membrane-cortex adhesion that couples the mechanics and hydrodynamics of the membrane to the force-dependent binding kinetics of the linker proteins. We predict the critical pressure difference that causes membrane-cortex detachment, and we discuss how cortical tension can be inferred from micropipette suction experiments. Then, we study the fluctuations of an adhered membrane, and suggest ways in which our predictions could allow probing membrane-cortex adhesion in fluctuation spectroscopy experiments. Then, we employ the proposed model to study the nucleation of blebs, which are balloon-like membrane protrusions arising from a local membrane-cortex detachment. We show that bleb nucleation is governed by membrane peeling, the fracture propagation process whereby adjacent membrane-cortex bonds break sequentially. Through this mechanism, bleb nucleation is not determined by the energy of a local detachment like in the classical nucleation picture, but rather by the kinetics of membrane-cortex linkers. We predict the critical radius for bleb nucleation through membrane peeling and the corresponding effective energy barrier. Finally, we simulate a fluctuating adhered membrane to obtain the probability distribution of bleb nucleation times. In Part II, we study the dynamics of active polar gels, which are soft materials, usually transiently-crosslinked polymeric networks, that are maintained out of equilibrium by internal processes that continuously transduce energy. We derive the constitutive equations of an active polar gel from a mesoscopic model for the dynamics of the molecules that crosslink the polar elements of the system. This way, we establish a connection between the molecular properties and the macroscopic behaviour of active polar gels. Specifically, we explicitly obtain the transport coefficients in terms of molecular parameters, showing that all transport coefficients have an active contribution that stems from breaking detailed balance for the crosslinker binding kinetics. In Part III, we study cell colonies and tissues, focusing in collective cell migration and tissue morphology. First, we propose a particle-based description of cell colonies to study how the different organizations of cells in tissues emerge from intercellular interactions. The model intends to capture generic cellular behaviours such as cell migration, adhesion, and cell-cell overlapping. In addition, it models the so-called contact inhibition of locomotion (CIL), which repolarizes cell migration away from cell-cell contacts, as a torque on the migration direction. We show how CIL yields an effective repulsion between cells, which allows to predict transitions between non-cohesive, cohesive, and 3D tissues. We conclude that, at low cell-cell adhesion, CIL hinders the formation of cohesive tissues. Yet, in continuous cell monolayers, CIL gives rise to self-organized collective motion, ensures tensile stresses in the monolayer, and opposes cell extrusion, thereby hindering the collapse of the monolayer into a 3D aggregate. Then, we focus on the spreading of epithelial monolayers, which we address by means of a continuum model based on the theory of active polar gels. First, we concentrate on the wetting transition of epithelial tissues, which separates monolayer spreading from retraction towards a 3D aggregate — namely the equivalent of a fluid droplet. We show that a critical radius exists for the wetting transition, which does not exist in the classical wetting picture. Thus, we show how the wetting properties of tissues emerge from active cellular forces, evidencing that the wetting transition has an active nature. Finally, we study the morphological stability of the front of a spreading monolayer. The model predicts that traction forces cause a long-wavelength instability of the monolayer front, whereas tissue contractility has a stabilizing effect. The predicted instability can explain the formation of finger-like multicellular protrusions observed during epithelial spreading. It can also explain the symmetry breaking of tissue shape observed during monolayer dewetting. By fitting the predictions to experimental data, we infer the monolayer viscosity and the noise intensity of tissue shape fluctuations, which we suggest to have an active origin.
 

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